Once defined, x may be noted a−1{\displaystyle a^{-1}}, where the fact that the inversion is m-modular is implicit.

The multiplicative inverse of a modulo m exists if and only ifa and m are coprime (i.e., if gcd(a, m) = 1).[1] If the modular multiplicative inverse of a modulo m exists, the operation of division by a modulo m can be defined as multiplying by the inverse of a, which is in essence the same concept as division in the field of reals.

Working in Z11{\displaystyle \mathbb {Z} _{11}}, one value of x that satisfies this congruence is 4 because

3⋅4=12≡1(mod11){\displaystyle 3\cdot 4=12\equiv 1{\pmod {11}}}

and there are no other values of x in Z11{\displaystyle \mathbb {Z} _{11}} that satisfy this congruence. Therefore, the modular multiplicative inverse of 3 modulo 11 is 4.

Once the inverse of 3 in Z11{\displaystyle \mathbb {Z} _{11}} is found, other values of x in Z{\displaystyle \mathbb {Z} } can be found that also satisfy the congruence. They may be obtained by adding multiples of 11 to the found inverse. Generalizing, all possible x for this example can be formed from

where a and b are given, and x, y and gcd(a, b) are the integers that the algorithm discovers. So, since the modular multiplicative inverse is the solution to

ax≡1(modm),{\displaystyle ax\equiv 1{\pmod {m}},}

by the definition of congruence, m | ax − 1, which means that m is a divisor of ax − 1. This, in turn, means that

ax−1=qm.{\displaystyle ax-1=qm.\,}

Rearranging produces

ax−qm=1,{\displaystyle ax-qm=1,\,}

with a and m given, x the inverse, and q an integer multiple that will be discarded. This is the exact form of equation that the extended Euclidean algorithm solves—the only difference being that gcd(a, m) = 1 is predetermined instead of discovered. Thus, a needs to be coprime to the modulus, or the inverse won't exist.

This algorithm runs in time O(log(m)2), assuming | a | < m, and is generally more efficient than exponentiation.

In the special case when m is a prime, φ(m) = m − 1 and the modular inverse is given by

a−1≡am−2(modm).{\displaystyle a^{-1}\equiv a^{m-2}{\pmod {m}}.}

This method is generally slower than the extended Euclidean algorithm, but is sometimes used when an implementation for modular exponentiation is already available. Some disadvantages of this method include:

The value φ(m) must be known, whose most efficient computation requires m's factorization. Factorization is widely believed to be a computationally hard problem. However, calculating φ(m) is straightforward when the prime factorisation of m is known.

The relative cost of exponentiation. Though it can be implemented more efficiently using modular exponentiation, when large values of m are involved this is most efficiently computed with the Montgomery reduction method. This algorithm itself requires a modular inverse mod m, which is what was to be calculated in the first place. Without the Montgomery method, we're left with standard binary exponentiation which requires division mod m at every step, a slow operation when m is large. Furthermore, any kind of modular exponentiation is a taxing operation.

The modular multiplicative inverse has many applications in algorithms, particularly those related to number theory, since many such algorithms rely heavily on the theory of modular arithmetic. As a simple example, consider the exact division problem where you have a list of odd word-sized numbers each divisible by k and you wish to divide them all by k. One solution is as follows:

Use the extended Euclidean algorithm to compute k−1, the modular multiplicative inverse of k mod 2w, where w is the number of bits in a word. This inverse will exist since the numbers are odd and the modulus has no odd factors.

For each number in the list, multiply it by k−1 and take the least significant word of the result.

On many machines, particularly those without hardware support for division, division is a slower operation than multiplication, so this approach can yield a considerable speedup. The first step is relatively slow but only needs to be done once.